2.1: Vectors

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Key Questions

• What differentiates a vector from a scalar?
• How do you identify the tip, tail, line of action, direction, and magnitude of any drawn vector?
• What are the standard notations for vectors and scalars in this textbook?
• What is the difference between the sense and orientation of a vector?

You can visualize a vector as an arrow pointing in a particular direction. The tip is the pointed end and the tail the trailing end. The tip and tail of a vector define a line of action. A line of action can be thought of as an invisible string along which a vector can slide. Sliding a vector along its line of action does not change its magnitude or its direction. Sliding a vector can be a handy way to simplify vector problems.

You can use this interactive diagram to adjust the magnitude and direction of the vector using point at the tip, the point of application of the vector by sliding the tail along the line of action, and the direction of the vector by moving the direction of the line of action.

Figure 2.1.1. Vector Definitions

The standard notation for a vector uses either an arrow above the vector name or the vector name in bold font. This book will use a bold font for vectors. You and your instructor will use an arrow above the vector name for handwritten work.

\[ \vec{F} = \vecarrow{F} = \text{ Vector } F \nonumber \]

Force vectors acting on physical objects have a point of application, which is the point at which the force is applied. Other vectors, such as moment vectors, are free vectors, which means that the point of application is not significant. Free vectors can be moved freely to any location as long as the magnitude and direction are maintained.

The vector’s magnitude is a positive real number including units which describes the ‘strength’ or ‘intensity’ of the vector. Graphically a vector’s magnitude is represented by the length of its vector arrow, and symbolically by enclosing the vector’s symbol with vertical bars. This is the same notation as for the absolute value of a number. The absolute value of a number and the magnitude of a vector can both be thought of as a distance from the origin, so the notation is appropriate. By convention the magnitude of a vector is also indicated, by the same letter as the vector, but in an non-bold font.

\[ F = |\vec{F}| = \text{magnitude of vector }\vec{F} \nonumber \]

By itself, a vector’s magnitude is a scalar quantity, but it makes no sense to speak of a vector with a negative magnitude so vector magnitudes are always positive or zero. Multiplying a vector by -1 produces a vector with the same magnitude but pointing in the opposite direction.

Vector directions are described with respect to a coordinate system. A coordinate system is an arbitrary reference system used to establish the origin and the primary directions. Distances are usually measured from the origin, and directions from a primary or reference direction. You are probably familiar with the Cartesian coordinate system with mutually perpendicular \(x\text{,}\) \(y\) and \(z\) axes and the origin at their intersection point.

Another way of describing a vector’s direction is to specify its orientation and sense. Orientation is the angle the vector’s line of action makes with a specified reference direction, and sense defines the direction the vector points along its line of action. A vector with a positive sense points towards the positive end of the reference axis and vice-versa. A vector representing an object’s weight has a vertical reference direction and downward sense or negative sense, for example.

A third way to represent a vector is with its unit vector multiplied by a scalar value called its scalar component. A unit vector is a vector with a length of one (unitless) which points in a defined direction. Hence, a unit vector represents pure direction, independent of the magnitude and unit of measurement. The scalar component is a signed number with units which may be positive or negative, and which defines the both the magnitude and sense of the vector. They should not be confused with vector magnitudes, which are always positive.

Vectors can either be either constant or vary as a function of time, position, or something else. For example, if a force varied with time according to the function \(F(t) = 10t\) [N] where \(t\) is the time in seconds, then the force would be \(\N{0}\) at \(t=0\text{,}\) and increase by \(\N{10}\) each second thereafter.